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\usepackage{amsmath, amsthm, amssymb, bm} % 数学公式与符号
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\usepackage{enumitem}

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%%文档的题目、作者与日期
\author{2024级数学与应用数学1班}
\title{常微分方程期末复习一}
\date{2025年12月2日}

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\begin{document}

\maketitle

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%\textbf{说明：} 以下题目覆盖常微分方程主要求解方法，请根据所学知识选择合适的方法求解或分析。

\begin{enumerate}[leftmargin=*, label=\textbf{\arabic*.}, itemsep=0.2cm]

\item \textbf{（分离变量法）} 求解初值问题：
$\frac{dy}{dx} = e^{x+y}, \quad y(0) = 0.
$

\item \textbf{（积分因子法）} 求解微分方程：
$
(2xy + y^2)\,dx + (x^2 + 2xy)\,dy = 0.
$
% 提示：验证是否为恰当方程；若否，尝试寻找仅依赖 $x$ 或 $y$ 的积分因子。

\item \textbf{（变量代换法）} 利用代换 $u = x + y$ 求解：
$
\frac{dy}{dx} = \sin(x + y).
$

\item \textbf{（微分法）} 求解微分方程：
$
y = x p - p^2, \quad \text{其中 } p = \frac{dy}{dx}. 
$
% 并指出其通解与奇解（包络）。

\item \textbf{（参数法）} 用参数表示法求解：
$
(y')^2 + 2y' - y = 0.
$
设 $p = y'$，将 $x$ 和 $y$ 表示为 $p$ 的函数。

\item \textbf{（幂级数方法）} 在 $x=0$ 附近求方程
$
y'' + x y' + y = 0
$
的幂级数解，写出前四项（至 $x^3$ 项）。

\item \textbf{（常数变易法）} 已知齐次方程 $y'' - 2y' + y = 0$ 的基本解组为 $\{e^x, x e^x\}$，用常数变易法求非齐次方程
$
y'' - 2y' + y = e^x \ln x \quad (x > 0)
$
的一个特解。%表达式（无需计算最终积分，但需写出变易后的方程组和特解形式）。

\item \textbf{（皮卡迭代法）} 对初值问题
$
\frac{dy}{dx} = x y, y(0) = 1,
$
写出前两个 Picard 迭代函数 $y_1(x)$ 和 $y_2(x)$. 

\item \textbf{（欧拉方法）} 将区间 $[0, 0.6]$ 三等分（即步长 $h = 0.2$），用欧拉方法近似求解初值问题
$
\frac{dy}{dx} = y - x, y(0) = 1,
$
并列出 $x = 0, 0.2, 0.4, 0.6$ 处的近似值。

\item \textbf{（特征方程法）} 求解常系数线性齐次微分方程：
$
y^{(4)} - y = 0.
$

\item \textbf{（待定系数法）} 求非齐次方程
$
y'' + y = x \cos x
$
的一个特解。%说明试探解的形式为何需包含 $x$ 的因子。

\item \textbf{（降阶法）} 求解不显含 $y$ 的二阶方程：
$
y'' = (y')^3.
$
提示：令 $p = y'$，则 $y'' = \frac{dp}{dx}$。

\item \textbf{（矩阵指数法）} 使用矩阵指数或特征值方法求解线性系统：
\[
\frac{d}{dt}\begin{bmatrix} x \\ y \end{bmatrix} = 
\begin{bmatrix} 3 & -1 \\ 1 & 1 \end{bmatrix}
\begin{bmatrix} x \\ y \end{bmatrix}.
\]

\item \textbf{（Frobenius 方法）} 考察方程
$
x^2 y'' + x y' + (x^2 - 1)y = 0
$
在 $x=0$ 附近的解。写出指标方程，并求出对应较大指标根的 Frobenius 级数解的前两项。

\item \textbf{（平衡点分析与线性化）} 考虑系统：
\[
\begin{cases}
\dot{x} = x(1 - x - y), \\
\dot{y} = y(0.5 - 0.25x - y).
\end{cases}
\]
求所有平衡点，并对每个平衡点进行线性化，判断其类型（结点、鞍点、焦点等）及稳定性。

% \item \textbf{（首次积分法）} 证明方程
% $
% y'' + \omega^2 y = 0 \quad (\omega > 0)
% $
% 具有首次积分 $E = \frac{1}{2}(y')^2 + \frac{1}{2}\omega^2 y^2$，并利用该守恒量求通解。

\end{enumerate}

\end{document}


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\begin{enumerate}
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\item  
求下列初值问题的解：$\frac{dy}{dx} = 1+y^2,\, y(x_0)=y_0$. 

\vspace{0.2cm}

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\item  
求解微分方程： $(ax+by)dx + (bx+cy)dy = 0$.   

\vspace{0.2cm}

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\item  
求解下列微分方程的初值问题：$xdx + ye^{-x}dy =0, \,\, y(0)=1$. 

\vspace{0.2cm}

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\item  
化为线性微分方程并求解：$\frac{dy}{dx} = \frac{y}{x+y^2}$. 

\vspace{0.2cm}

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\item  
利用适当的变换，求解微分方程： $y'=\cos(x-y)$. 

\vspace{0.2cm}

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\item  

求初值问题 $\frac{dy}{dx} = x+y+1,\,\, y(0)=0$ 的皮卡序列，并由此求极限求解。

\vspace{0.2cm}

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\item  

将区间 $[0,1]$ 三等分，求初值问题 $\frac{dy}{dx} = x+y,\,\, y(0)=0$ 的欧拉折线。写出分段函数的表达式。

\vspace{0.2cm}

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\item  
讨论下列微分方程的解的存在区间：
$\frac{dy}{dx} = \frac{1}{x^2+y^2}, y(0)=0$. 

\vspace{0.2cm}

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\item 
用微分法求解微分方程$2y = p^2 +4px +2x^2$, 其中 $p=\frac{dy}{dx}$. 

\vspace{0.2cm}

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\item  用参数法求解微分方程 $2y^2 +5p^2=4$, 其中 $p=\frac{dy}{dx}$. 

\vspace{0.2cm}

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\item  
求常系数非齐次线性微分方程组 $\frac{d \vec{y}}{dx}=A\vec{y} + \vec{f}(x)$ 的通解，其中 
$A = \begin{bmatrix} 2 & 1 \\ 0 & 2 \end{bmatrix},\,\, \vec{f}(x) = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$. 

\vspace{0.2cm}

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\item  
求解下列常系数线性微分方程：
$y''' + 3y' -4y = 0$.

\vspace{0.2cm}

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\item  
求出微分方程在 $x_0$ 附近的两个线性无关的幂级数解，
$(1-x)y'' + y = 0, \,\, x_0=0$. 

\vspace{0.2cm}

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\item  研究二维微分方程组的两个平衡点的稳定性：
\begin{align*}
dx/dt &= y, \\ 
dy/dt &= -1+x^2. 
\end{align*}

\vspace{0.2cm}

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\item 
判断下列方程的奇点 $(0,0)$ 的类型，并作出该奇点附近的相图：
\begin{align*}
dx/dt &= 2x+4y+\sin y, \\ 
dy/dt &= x+y+e^y-1. 
\end{align*}

\vspace{0.2cm}

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\end{enumerate}



